This talk has three parts. First, I want to explain how monoid acts can be viewed as models of computation. Second, I will sketch my thesis work on categories of actions of topological monoids. Third, if there is time, I would like to explain how this work would need to be extended in order to capture the monoid acts from the first part, point out some obstacles that need to be overcome, and give some intuition on what building such a topos-theoretic context could provide. Note that while I will give complexity-theoretic motivation, I want to apply this theory to monoid and group acts of all kinds, so feel free to ask questions about your favourite ones.